\( t_s =\frac{2 n_1 \cos (\theta_1 ) }{n_1 \cos (\theta_1 ) + n_2 \cos (\theta_2 )} \)
\( r_s =\frac{ n_1 \cos (\theta_1) - n_2 \cos (\theta_2 )}{n_1 \cos (\theta_1 ) + n_2 \cos (\theta_2 )} \)
\( t_p =\frac{2 n_1 \cos (\theta_1 ) }{n_1 \cos (\theta_2 ) + n_2 \cos (\theta_1 )} \)
\( r_p =\frac{ n_1 \cos (\theta_2) - n_2 \cos (\theta_1 )}{n_1 \cos (\theta_2 ) + n_2 \cos (\theta_1 )} \)
\(\theta_i = \theta_r \)
\(n_1 \sin ( \theta_1 ) = n_2 \sin ( \theta_2 )\)
Vector field demonstrating the magnetic forces applied in a space
Scalar quantity \(\Phi\) representing the total magnetic field through the surface area of a material
\(\Phi = \iint_{S}\textbf{B} \cdot d\textbf{S} \)
The charge of a single proton or electron in coulombs
\(1.602176634 \times 10^{-19}\text{C}\)
Scalar quantity \(q\) representing the physical property in material's matter that causes a force when exposed to an electric field. An electron holds a charge \(-1e\), while a proton holds a charge \(+1e\)
\( q(t) = \int_{0}^{t} I(t') dt'\)
Denoted as \(e\), represent the smallest possible charge, equivalent to \(1e=1.6 \times 10^{-19}C\) in Coulombs
Charges per unit of volume
\(\rho = \frac{Q_{\text{enc}}}{V}\)
Position of lower charge from which electrons flow out of
Position of higher charge from which electrons flow towards
Vector field demonstrating the forces applied to charged particles
Scalar quantity \(I\) representing the rate at which charge flows in a material from the anode to the cathode
\( I(t) = \frac{dq}{dt} \)
Current is the flow of electrons from a place of lower charge (anode) to a place of higher charge (cathode)
Arbitrarily defining the flow of current from cathode to anode for notational purposes
Vector representing the direction and magnitude of charge flow per a unit of area
\(\textbf{J}= \frac{I_{\text{enc}}}{A}\)
Scalar quantity \( V \) representing the work required to move one single charge from a reference point to a certain point in an electric field. This manifests the phenomenon of points with higher charges requiring less energy to move to.
\(\displaystyle V (\textbf{r}_1) = - \int_{\mathcal{C}} \textbf{E} \cdot d\textbf{r} \)
Scalar quantity \( U \) representing the work required to move a charge of magnitude \(q\) from a reference point to a certain point in an electric field.
\(U(\textbf{r}_1) = q V(\textbf{r}_1) \)
Scalar quantity \( \Delta V \) representing the difference of electric potential energy of two points; comparing two points with eachother rather than comparing them both with the reference point
Since like charges repel and dislike charges attract, the difference in charge causes current to occur.
\( \Delta V(\textbf{r}_1 , \textbf{r}_2) = V( \textbf{r}_2 ) - V( \textbf{r}_1 ) \)
Notationally, \( \Delta V \) can be shortened to \(V\) (since \( \textbf{r}_1 \text{ is the chosen reference point } \implies \Delta V(\textbf{r}_1 , \textbf{r}_2) = V(\textbf{r}_2 )\) )
Scalar quantity \(R\) representing a material's resistance to current
\(R(t) = \frac{V (\textbf{r}(t))}{I(t)}\)
\(R = \frac{\rho \ell}{A}\)
Scalar quantity \(G\) representing a material's propensity to conduct changes throughout itself
\(G = \frac{1}{R}\)
Empirical law stating proportionality between current and voltage for ohmic materials. This implies the corrolary that resistance is constant in ohmic materials
\(V \propto I\)
Scalar quantity \(E\) representing energy created by current that can be transferred into work
\( E=qV(\textbf{r}_1) \)
Scalar quantity \(P\) representing rate at which energy is transferred
\( P = \frac{dE}{dt}\)
\( P= \frac{dE}{dt}= \frac{dq}{dt}V=IV \)
\( P=IV=I^{2}R=\frac{V^2}{R} \)
\( E= \int_{t_1}^{t_2} P(t) dt \)
Law asserting conservation of charge in a circuit; charge cannot be lost so therefore all current flowing into a node must flow out of the node
\( \sum I_{n} = 0 \)
Law asserting conservation of energy in a circuit; energy cannot be lost so therefore the voltage drops around a closed circuit sum to the voltage of the closed circuit
This means voltage is uniform in parallel from the electrically common points
\(\sum V_{m} = 0\)
\( R_{eq} = \sum R_{m}\)
\( V = \sum V_{m} = I(\sum R_{m}) = I R_{eq} \)
\( \frac{1}{R_{eq}} = \sum \frac{1}{R_{m}}\)
\( I = \sum I_{n} = V(\sum \frac{1}{R_{n}}) = V(\frac{1}{R_{eq}}) \)
In parallel, the current is shared between each branch according to KCL. Using KCL, Ohm's law, and the fact that voltage across a parallel circuit is uniformi:
\( I_{m} =\frac{R_{eq}}{R_{m}}I \)
\(I_{m} =\frac{V}{R_{m}} = \frac{IR_{eq}}{R_{m}}\)
In series, the voltage is shared between each element. Using KVL, and Ohm's law, and the fact that current across a series circuit is uniform:
Scalar quantity \(C\) representing material's ability to store electric charges per volt
\( C = \frac{q}{V}\)
\( C = \frac{\varepsilon A}{\ell}\)
\( i(t)= C\frac{dV(t)}{dt}\) where:
\( V(t) = \frac{1}{C} \int_{t_{0}}^{t} I(t) dt + V(t_{0}) \) where:
\( W_{c}(t) = \frac{CV^2(t)}{2} \) where:
Quantity \(\varepsilon\) that describes how easily a material polarises with an applied voltage, how easily polarizing is permitted to occur
Permittivity of a vacuumm denoted \(\varepsilon_0\)
\(\varepsilon_{0} = 8.8541878128(13) \times 10^{-12} F/m\)
\( \frac{1}{C_{eq}} = \sum \frac{1}{C_{n}}\)
\( C_{eq} = \sum C_{n}\)
\( I_{m} = C_{m} \frac{dv_{c}}{dt} \)
\( I_{T} = \frac{dv_{c}}{dt} \sum C_{m} \) (KCL)
\( \therefore C_{eq} = \sum C_{n}\)
When conducting transient analysis on capacitors, the coefficient for time can be deduced using knowlegde of differential equations and the nature of circuits to be the following:
\(\tau = RC \)
The following shows how fast the current in an inductor changes with respect to time
\(I_{c} = \frac{V_{b}}{R} (e^{-\frac{t}{\tau}}) \)
\(V_{c} = V_{b} (1-e^{-\frac{t}{\tau}}) \) where:
And by using the fact \(I = C\frac{dV}{dt}\), we see how the voltage changes in respect to time
This means during charging, current fades and voltage accumulates in a capcitor
\(I_{c} = -\frac{V_{Cmax}}{R} e^{-\frac{t}{\tau}} \)
\(V_{c} = V_{Cmax}e^{-\frac{t}{\tau}} \) where:
And by using the fact \(V = L\frac{dI}{dt}\), we see how the voltage changes in respect to time
This means during discharging, current in the opposite direction is jolted and fades and voltage fades in a capcitor
Phenomenon of change in a magnetic field across an electrical conductor producing a voltage called Electromotive Force (EMF). This is the basis of Faraday's law, and relates the fields of magnetism and electricity.
Scalar quantity \(L\) representing a material's ability to form a magnetic field per ampere
\( L = \frac{\lambda}{I} \)
\( L \equiv N^2 \frac{\mu A}{\ell}\)
\( V = L\frac{dI(t)}{dt}\) where:
\( I(t) = \frac{1}{L} \int_{t_{0}}^{t} V(t) dt + I(t_{0}) \) where:
\( W_{L}(t) = \frac{Li^2(t)}{2} \) where:
When a current source is removed from a circuit, inductors release an Electromagnetic frequency that creates a bit of voltage to oppose this change. This makes
Quantity \(\mu \) that describes the amount of magnetisation in an object when a magnetic field is applied to it, how much a magnetic field permeates in a material
Permeability of a vacuumm denoted \(\mu_0\)
\(\mu_{0} = 4\pi \times 10^{-7} H/m\)
\( L_{eq} = \sum L_{n}\)
\( \frac{1}{L_{eq}} = \sum \frac{1}{L_{n}}\)
When conducting transient analysis on inductors, the coefficient for time can be deduced using knowlegde of differential equations and the nature of circuits to be the following:
\(\tau = \frac{L}{R} \)
The following shows how fast the current in an inductor changes with respect to time
\(I_{L} = \frac{V_{b}}{R} (1-e^{-\frac{t}{\tau}}) \) where:
\(V_{L} = V_{b} e^{-\frac{t}{\tau}} \) where:
And by using the fact \(V = L\frac{dI}{dt}\), we see how the voltage changes in respect to time
\(I_{L} = I_{Lmax} e^{-\frac{t}{\tau}} \)
\(V_{L} = I_{Lmax} R e^{-\frac{t}{\tau}} \) where:
And by using the fact \(V = L\frac{dI}{dt}\), we see how the voltage changes in respect to time
Inverse square law representing the electric field of one particle centered at the origin in spherical coordinates. This representation of an electric field is the base from which Maxwell's equations are derived.
\( \textbf{E} = \frac{Q}{4 \pi \varepsilon_{0} \rho^2} \hat{\rho} \)
An electric charge of density \(q\) is therefore subject to the following force:
\( \textbf{F}= q\textbf{E} \)
Electric fields representing the superposition of multiple electric charges can be accounted for by summing the electric fields of each electric charge
Law representing the magnetic field of a current travelling up the z-axis in cylindrical coordinates. This representation of a magnetic field is the base from which Maxwell's equations are derived.
\( \textbf{B} = \frac{\mu_0 Q}{4 \pi \rho} \hat{\theta} \)
Set of differential equations governing classical electromagnetism by stating the properties of electic and magnetic fields that are either derived from Coulomb's law and Biot-Savart law and an empirical observation (Faraday's law).
Law stating that the divergence of the electric flux density is the electric charge density
\( \nabla \cdot \textbf{E} = \frac{\rho(\textbf{r}(t)}{\varepsilon_0} \)
\( \displaystyle \iint_{S} \textbf{E} \cdot d\textbf{S} = \frac{Q_{\text{enc}}}{\varepsilon_0} \)
Law stating magnetic fields have zero divergence, they are conserved under flux
\( \nabla \cdot B = 0 \)
\( \displaystyle \iint_{S} \textbf{B} \cdot d\textbf{S} = 0 \)
Law stating that the change in a magnetic field produces a change in an electric field's curl (work to move an electric charge around electric field) of opposite polarity
Law stating that electromotive force (curl of electric field) equals the magnetic field induced by
Note that using the Coulomb's law and taking the curl returns \(0\), however this experimental law makes the mathematical structure more realistic
\(\nabla \times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t} \)
\(\displaystyle \oint_{\mathcal{\partial S}} \textbf{E} \cdot d\textbf{r} = - \iint_{S} \frac{\partial \textbf{B}}{\partial t} \cdot d\textbf{S}\)
Law stating that the curl of magnetic field
\( \nabla \times \textbf{B} = \mu_0 \textbf{J}(\textbf{r}(t)) + \mu_0 \varepsilon_0 \frac{\partial \textbf{E}}{\partial t}\)
\( \displaystyle \oint_{\mathcal{C}} \textbf{B} \cdot d\textbf{r} = \mu_0 I_{\text{enc}} \)
Law asserting the force of a charged particle with velocity \(\textbf{v}\) amidst an electric and magnetic field
\(\textbf{F} = q(\textbf{E} + \textbf{v} \times \textbf{B})\)
Tendency of current density to be maximised on a surface and exponentially decaying approaching the center
Current with magnitude oscillating between directions, modelled by sinusoidal functions
Current with constant magnitude